Encounter with a Geometer, Part I, Volume 47, Number 2

T he aim of this article is to communicate the work of Mikhael Gromov (MG) and its influence in almost all branches of contemporary mathematics and, with a leap of faith, of future mathematics. Because of its length, the article will appear in two parts. It is not meant to be a technical report, and, in order to make it accessible to a wide audience, I have made some difficult choices by highlighting only a few of the many subjects studied by MG. In this way I can be more leisurely in my exposition and give full definitions, results, and even occasional hints of proofs. I wrote this article because I believe that MG’s work is greatly underrated. I will not analyze the reasons for this phenomenon, although a general idea of why this is so will become clear from a reading of the text. Apart from Riemannian geometry, notable exceptions to this phenomenon occur in two fields: hyperbolic groups and symplectic geometry via pseudoholomorphic curves; these areas will be discussed in the first two sections below. In any case, MG expressed his own view to me: